Computational Geosciences

, Volume 22, Issue 1, pp 125–144 | Cite as

Surrogate combining harmonic decomposition and polynomial chaos for seismic shear waves in uncertain media

Original Paper


A polynomial chaos (PC) surrogate is proposed to reconstruct seismic time series in one-dimensional (1D) uncertain media. Our approach overcomes the deterioration of the PC convergence rate during long time integration. It is based on a double decomposition of the signal: a damped harmonic decomposition combined with a polynomial chaos expansion of the four coefficients of each harmonic term (amplitude, decay constant, pulsation, and phase). These PC expansions are obtained through the least squares method which requires the solution of nonlinear least squares problems for each sample point of the stochastic domain. The use of the surrogate is illustrated on vertically incident plane waves traveling in 1D layered, vertically stratified, isotropic, viscoelastic soil structure with uncertainties in the geological data (geometry, wave velocities, quality factors). Computational tests show that the stochastic coefficients can be efficiently represented with a low-order PC expansion involving few evaluations of the direct model. For the test cases, a global sensitivity analysis is performed in time and frequency domains to investigate the relative impact of the random parameters.


Uncertainty propagation Elastodynamic wave equation Pseudo-harmonic decomposition Non intrusive spectral method Least squares Global sensitivity analysis 


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This work is supported by internal fundings of the BRGM (French Geological Survey). P. Sochala is greatful to O. Le Maître for his constructive comments on the polynomial chaos expansion and to F. Smaï for fruitful discussions concerning the harmonic inversion problem. F. De Martin is thankful to F. Hollender, E. Chaljub, E. Maufroy, and P.-Y. Bard for providing the data used for the double layer test case and for their contributions to determine the distributions of the uncertain parameters. The Thomson–Haskell propapagator matrix computations have been done on the 32-cores Intel ®; cluster provided by P. Thierry.


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.BRGMOrléansFrance

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